## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 10, Number 2 (2016), 2641-2684.

### Exact asymptotics for the scan statistic and fast alternatives

James Sharpnack and Ery Arias-Castro

#### Abstract

We consider the problem of detecting a rectangle of activation in a grid of sensors in $d$-dimensions with noisy measurements. This has applications to massive surveillance projects and anomaly detection in large datasets in which one detects anomalously high measurements over rectangular regions, or more generally, blobs. Recently, the asymptotic distribution of a multiscale scan statistic was established in [18] under the null hypothesis, using non-constant boundary crossing probabilities for locally-stationary Gaussian random fields derived in [8]. Using a similar approach, we derive the exact asymptotic level and power of four variants of the scan statistic: an oracle scan that knows the dimensions of the activation rectangle; the multiscale scan statistic just mentioned; the adaptive variant; and an $\epsilon $-net approximation to the latter, in the spirit of [3]. This approximate scan runs in time near-linear in the size of the grid and achieves the same asymptotic level and power as the adaptive scan, and has a poly-logarithmic time parallel implementation. We complement our theory with some numerical experiments, and make some practical recommendations.

#### Article information

**Source**

Electron. J. Statist., Volume 10, Number 2 (2016), 2641-2684.

**Dates**

Received: December 2015

First available in Project Euclid: 12 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1473685450

**Digital Object Identifier**

doi:10.1214/16-EJS1188

**Mathematical Reviews number (MathSciNet)**

MR3546971

**Zentralblatt MATH identifier**

1345.62078

**Subjects**

Primary: 62G10: Hypothesis testing 62M40: Random fields; image analysis 60G32

**Keywords**

Sensor networks image processing multiscale detection scan statistic suprema of Gaussian random fields

#### Citation

Sharpnack, James; Arias-Castro, Ery. Exact asymptotics for the scan statistic and fast alternatives. Electron. J. Statist. 10 (2016), no. 2, 2641--2684. doi:10.1214/16-EJS1188. https://projecteuclid.org/euclid.ejs/1473685450